Given expression is (1+x)101(1−x+x2)100 =(1+x)[(1+x)(1−x+x2)]100 =(1+x)(1+x3)100 So, coefficient of x50 in (1+x)101(1−x+x2)100 = coefficient of x50 in (1+x)(1+x3)100 = coefficient of x50 in (1+x3)100 + coefficient of x49 in (1+x3)100 ∵ In the expansion of (1+x3)100, the power of x is multiple of 3, and while 50 and 49 are not multiple of 3. So, coefficient of x50 and 49 in the expansion of (1+x3)100 is zero. ∴ Coefficient of x50 in (1+x)100(1−x+x2)101 is zero.